Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24.
TAKING CATEGORIES SERIOUSLY
F. WILLIAM LAWVERE
Abstract. The relation between teaching and research is partly embodied in simple
general concepts which can guide the elaboration of examples in both. Notions and constructions, such as the spectral analysis of dynamical systems, have important aspects
that can be understood and pursued without the complication of limiting the models to
specific classical categories. Pursuing that idea leads to a dynamical objectification of
Dedekind reals which is particularly suited to the simple identification of metric spaces
as enriched categories over a special closed category. Rejecting the complacent description of that identification as a mere analogy or amusement, its relentless pursuit [8] is
continued, revealing convexity and geodesics as concepts having a definite meaning over
any closed category. Along the way various hopefully enlightening exercises for students
(and possible directions for research) are inevitably encountered: (1) an explicit treatment of the contrast between multiplication and divisibility that, in inexorable functorial
fashion, mutates into the adjoint relation between autonomous and non-autonomous dynamical systems; (2) the role of commutation relations in the contrast between equilibria
and orbits, as well as in qualitative distinctions between extensions of Heyting logic; (3)
the functorial contrast between translations and rotations (as appropriately defined) in
an arbitrary non symmetric metric space.
The theory of categories originated [1] with the need to guide complicated calculations
involving passage to the limit in the study of the qualitative leap from spaces to homotopical/homological objects. Since then it is still actively used for those problems but also
in algebraic geometry [2], logic and set theory [3], model theory [4], functional analysis
[5], continuum physics [6], combinatorics [7], etc. In all these the categorical concept of
adjoint functor has come to play a key role.
Such a universal instrument for guiding the learning, development, and use of advanced
mathematics does not fail to have its indications also in areas of school and college mathematics, in the most basic relationships of space and quantity and the calculations based on
those relationships. In saying “take categories seriously”, I advocate noticing, cultivating,
After the original manuscript was stolen, I was able to reconstruct, in time for inclusion in the
proceedings of the 1983 Bogotá Workshop, this“sketch” (as Saunders Mac Lane’s review was to describe
it). Interested teachers can elaborate this and similar material into texts for beginning students. For
the present opportunity, I am grateful to Xavier Caicedo who gave permission to reprint, to Christoph
Schubert for his expert transcription into TEX, and to the editors of TAC.
Received by the editors 2004-11-02.
Transmitted by M. Barr, R. Rosebrugh and R. F. C. Walters. Reprint published on 2005-03-23.
2000 Mathematics Subject Classification: 06D20, 18D23, 37B55, 34L05, 52A99, 53C22, 54E25, 54E40.
Key words and phrases: General spectral theory, Nonautonomous dynamical systems, Symmetric
monoidal categories, Semimetric spaces, General convexity, Geodesics, Heyting algebras, Special maps
on metric spaces.
c ReThe article originally appeared in Revista Colombiana de Matemáticas, XX (1986) 147-178. vista Colombiana de Matemáticas, 1986, used by permission.
1
2
F. WILLIAM LAWVERE
and teaching of helpful examples of an elementary nature.
1. Elementary mutability of dynamical systems
Already in [1] it was pointed out that a preordered set is just a category with at most one
morphism between any given pair of objects, and that functors between two such categories are just order-preserving maps; at the opposite extreme, a monoid is just a category
with exactly one object, and functors between two such categories are just homomorphisms
of monoids. But category theory does not rest content with mere classification in the spirit
of Wolffian metaphysics (although a few of its practitioners may do so); rather it is the
mutability of mathematically precise structures (by morphisms) which is the essential
content of category theory. If the structures are themselves categories, this mutability is
expressed by functors, while if the structures are functors, the mutability is expressed by
natural transformations. Thus if Λ is a preordered set and X is any category (for example the category of sets and mappings, the category of topological spaces and continuous
mappings, the category of linear spaces and linear transformations, or the category of
bornological linear spaces and bounded linear transformations) then there are functors
Λ −→ X
sometimes called “direct systems” in X , and the natural transformations
Λ
'
7X
between two such functors are the appropriate morphisms for the study of such direct
systems as objects.
An important special case is that where Λ = 0 → 1 , the ordinal number 2, then the
functors 2 −→ X may be identified with the morphisms in the category X itself; likewise
if Λ = 0 → 1 → 2 → · · · is the ordinal number ω, functors
X
ω −−→ X
are just sequences of objects and morphisms
X0 −→ X1 −→ X2 −→ X3 −→ . . .
f
fn
in X , and a natural transformation X −−→ Y between two such is a sequence Xn −−→ Yn
of morphisms in X for which all squares
Xn
fn
Xn+1
fn+1
/ Yn
/ Yn+1
TAKING CATEGORIES SERIOUSLY
3
commute in X (here the vertical maps are the ones given as part of the structure of X
and Y ).
Similarly, if M is a monoid then the functors M −→ X are extremely important mathematical objects often known as actions of M on objects of X (or representations of M
by X -endomorphisms, or . . . ) and the natural transformations between such actions are
known variously as M -equivariant maps, intertwining operators, homogeneous functions,
etc. depending on the traditions of various contexts. Historically the notion of monoid (or
of group in particular) was abstracted from the actions, a pivotally important abstraction
since as soon as a particular action is constructed or noticed, the demands of learning,
development, and use mutate it into: 1) other actions on the same object, 2) actions on
X
other related objects, and 3) actions of related monoids. For if M −−→ X is an action and
h
M −−→ M is a homomorphism, then (composition of functors!) Xh is an action of M ,
C
while if X −−→ Y is a functor, then CX is an action of M on objects of Y. To exemplify,
if M is the additive group of time-translations, then a functor M −→ X is often called
a dynamical system (continuous-time and autonomous) in X , but if we are interested in
h
observing the system only on a daily basis we could consider a homomorphism M −−→ M
where M = N is the additive monoid of natural numbers, and concentrate attention on
the predictions of the discrete-time, autonomous, future-directed dynamical system Xh.
In other applications we might have M = M = the multiplicative monoid of real num( )p
bers, but consider the homomorphism M −−−→ M of raising to the pth power; then if we
f
//
are given two actions M
X on objects of X , a natural transformation X −−→ (Y )p is
just a morphism of the underlying X -objects which satisfies
f (λx) = λp f (x)
x
for all λ in M and all T −−→ X in X , i.e. a function homogeneous of degree p. An
extremely important example of the second mutation of action mentioned above is that
in which Y is the opposite of an appropriate category of algebras and the functor C
assigns to each object (domain of variation) of X an algebra of functions (= intensive
quantities) on it. Then the induced action CX of M describes the evolution of intensive
quantities which results from the evolution of “states” as described by the action X.
A frequently-occurring example of the third type of mutation of action arises from the
surjective homomorphisms M −→ M from the additive monoid of time-translations M X
to the circle group M . Then a dynamical system M −−→ X is said to be “periodic of
period h” if there exists a commutative diagram of functors as follows:
M CC
CC
CC
h CC!
X
M
/X
>
4
F. WILLIAM LAWVERE
2. Kan quantifiers in spectral analysis
Most dynamical systems are only partly periodic, and such an analysis can conveniently
be expressed by “Kan-extensions” as follows (we do not assume that M , M are monoids):
h
M
M
For a functor M −−→ M and acategory X , the induced functor X
will
−→ X
often have a left adjoint X −→ h X and a right adjoint X −→ h X. These Kan
adjoints vastly generalize the existential and universal quantifiers of logic, special cases
arising when all the objects of X are truth-values. (Usually this just means that the
objects of X are canonically idempotent with respect to cartesian product or coproduct.)
Kan adjoints also generalize the induced representations frequently considered when X is a
linear category (i.e. finite coproduct is canonically isomorphic to finite product) and when
M−→ M is of “finite index”, in which case there is a strong tendency for h ( ) and
h ( ) to coincide. The defining property of these as adjoints are the natural bijections
T −→ h X
h X −→ Y
X −→ Y h
T h −→ X
between M -natural, respectively M -natural transformations, where Y , T are functors
M −→ X (that is objects of the category X M whose morphisms are the M -natural
transformations) and X is a functor M −→ X (that is an object of the category X M
whose morphisms are M -natural transformations). Since these refined “rules of inferk
ence” uniquely characterize the adjoints up to unique natural isomorphism, if M −−→
h
M −−→ M are two functors for each of which the two Kan adjoints exist, then from the
associativity of substitution, Y (hk) = (Y h)k, follow the two rules
(hk) Z ∼
= h (k Z)
(hk) Z ∼
= h (k Z).
If M is a discrete category I with I objects (and no morphisms except the identity
morphisms) and if M is the single morphism category 1, then there is a unique functor
I −→ 1, often also called I and the Kan adjoints are just the coproduct and product
functors respectively:
I X = i∈I Xi
I X = i∈I Xi
X
where a functor I −−→ X is just a family of objects. It is chiefly in regard to the
existence of Kan extensions that questions of “largeness”
and “smallness”
enter category
theory. The class of all categories X for which h ( ) and h ( ) exist in X can be
h
called the “smallness” of M −−→ M , while dually (in the sense of Galois connections)
the class of all functors h for which these exist over a given X can be called the degree of
“(bi) completeness” of X , with obvious refinements for left completeness where only
is considered and for right completeness where only
is considered. Informally we may
5
TAKING CATEGORIES SERIOUSLY
h
just say that M −−→ M is sufficiently small for X or that X is sufficiently complete for
h when these constructions can be carried out.
h
Returning to the example of a “period”, i.e. a surjective homomorphism M −−→ M
from the additive group of time-translations to the circle group, the induced functor
X M → X M
is just the full inclusion, into the category of all X -dynamical systems (continuous, autonomous),
of the subcategory of those that happen to have period h. Then the construction h X just gives the part of X consisting of the h-periodic states. More
precisely
the following adjunction morphism (derived from the rule of inference for )
(h
X)h −→ X
will typically be the inclusion of the h-periodic part.
[Of course there is also
X −→ (h X)h
obtained by forcing the arbitrary dynamical system X into the h-periodic mold, with
an accompanying collapse of states, whose detailed understanding depends on a detailed
understanding of the “collapsing” or quotient process in X . The quotient process is just in
/
o
/ V in which the two composites
( ). Here ∆op
general ∆op
1
1 is the finite category E
at V are both the identity (implying that the two composites at E are idempotents which
absorb one another in a non-commutative way) and functors ∆op
1 −→ X are often referred
to as (reflexive) graphs in X ; ∆1 itself can be concretely represented as the full category
of the category of categories consisting of the two objects V = 1 and E = 2 = 0 → 1 in
//
which representation the two arrows V
E in ∆1 are the two adjoints of the unique
E −→ V . The reflexive graph in X arising from a period h (homomorphism of monoids)
and a particular dynamical system X is just X̌ given by
/
Eh · Xf
/X
where Eh is the set of all pairs m1 , m2 for which h(m1 ) =h(m2 ) and Eh · X is the
X̌ depend sensitively on
coproduct of Eh copies of X. The detailed properties of ∆op
1
the nature of the category X , usually inconcrete examples more so than do the detailed
properties of the dual construction ∆1 X̂, where X̂ is the ∆1 −→ X given by
Xe
/
/ X Eh
(where X Eh denotes the product of Eh copies of X, and where, as throughout this bracket,
we have followed the usual abuse of notation of using the same letter X to denote
also
theobject X(0) of X that underlies the action of the monoid M ); thus ∆1 X̂ is the
(h X)h outside this bracket.]
6
F. WILLIAM LAWVERE
X
The full period spectrum of a dynamical system M −−→ X can be regarded as a
single functor as follows. Say that a period h2 divides a period h1 if there exists an
endomorphism q of the circle group M such that
M CC
{
CC h2
h1 {{
CC
{{
CC
{
}{{
!
q
/M
M
h2 = qh1
(then q is unique because h1 is assumed surjective and q itself is surjective because h2 is
assumed to be). Denote by Q the category (actually a pre-ordered set) whose objects are
the periods and whose morphisms are the q as indicated. By the definitions, if h2 divides
h1 and if X is (part of) a dynamical system of period h2 , then it is (part of) a dynamical
system of period h1 as well:
M ?
M

h1 




X
/5 X
@
?? Y1
??
??
??h2
/M
∃Y2 =⇒ ∃Y1
Y2
Similar reasoning shows that for any dynamical system X, q induces in a functorial way
an inclusion
(h2 X)h2 −→ (h1 X)h1
of the h2 -periodic part of X into the h1 -periodic part of X, whenever q is the
reason for
op
h2 dividing h1 . Thus we get a functor X̄ : Q −→ X (where X̄(h) = (h X)h) that
in turn depends functorially on X so that X −→ X̄ defines the “periodic pre-spectrum”
functor
(¯)
X M −−→ X Q
from dynamical systems in any sufficiently -complete category X into the category of
direct systems in X indexed by the poset Q of periods.
op
More closely corresponding to the usual notion of spectrum
is the following: the weight
attached to a given period h is not so much the space (h X)h of states having that period
as it is the smaller space of orbits of such states, where in general the notion of orbit space
is the left adjoint
M
()
X M −−−−−−→ X
to the functor induced by the unique M −→ 1. Since for any h, the space (h X)h of
h-periodic points is itself a dynamical system in its own right, hence combining these as
h varies through Q we get a lifted pre-spectrum functor indicated by the dotted arrow
below; the latter can be composed with the functor induced by the orbit space functor
TAKING CATEGORIES SERIOUSLY
7
(upon parameterizing by Q its inputs and outputs):
(¯)
X
M
/ (X M )Qop (M
JJ
JJ
JJ
J
(¯) JJJ%
%
/ X Qop
( ))Q
XQ
op
¯ . The periodic spectrum X̄
¯
to yield what could be called the periodic spectrum X −→ X̄
of a dynamical system X can in many cases be pictured as
Q
h
h
where thedarkness
of the line at a period h ∈ Q is proportional to the size of the
space M ((h X)h) of equivalence classes of states of period h, where two states are
equivalent if the dynamical action moves one through the other.
There is one other point (not in Q) which may also be
considered part of the spectrum,
X is the space of fixed states
namely the map M −→ 1 whose corresponding M
of the dynamical system X. If M is a group, then the fixed point space is usually
a subspace of the orbit space, for example if X is the category of sets and mappings.
The same conclusion follows if M is any commutative monoid. However, for the three
element monoid (essentially equivalent, insofar as actions are concerned, to the category
∆1 mentioned above) consisting of the morphisms 1, δ0 , δ1 with the multiplication table
X
δi δj = δi one can find examples of (right) actions ∆op
1 −−→ S on sets (i.e. reflexive graphs)
having
of fixed points (i.e. vertices) but only one orbit so that the map
any givenopnumber
X
−→
∆
X
is
not at all a monomorphism.
∆op
1
1
3. Enhanced algebraic structure in dynamics and logic
Incidentally, the above remark that both groups and commutative monoids share a property not true for all monoids can be made more explicitly algebraic by the following
exercise. If C is a category that is either a group, i.e. every morphism in C has a two-sided
inverse, or a commutative monoid, then C acts on its endomorphisms in the following
a
way: for any morphism X −−→ X in C, and for any endomorphism x of X, we can define
an endomorphism xa of X such that
xa = axa
X
xa
a
/X
x
8
F. WILLIAM LAWVERE
and moreover this is an action in the sense that 1a = 1 and
xab = (xa )b
X b
/ X
a
/X
x
and even an action by monoid homomorphisms in that
(xy)a = xa y a
1a = 1
for any two endomorphisms x, y of the codomain of a. Of course if C is itself a monoid
then all its morphisms are endomorphisms, and if all morphisms in C are monomorphisms
(a cancellation law) then there could be at most one operation x, a −→ xa with the crucial
property xa = axa . In the intersection of the two claimed cases, (i.e. for abelian groups)
the two formulas for xa reduce to the same (trivial) thing. If we restrict consideration to
the full subcategory of monoids determined by groups and commutative monoids (i.e. the
union of the two kinds of objects but containing all four types of homomorphisms between
them) we get an example of a natural operation on the underlying-set functor that does
not extend (from the full subcategory) to all monoids; note that the “algebraic structure”
of a full subcategory of an equationally defined algebraic category may have additional
operations as well as additional identities between the given operations. In our example
the subcategory is the union (made full) of two full subcategories which are themselves
equationally defined (in the sense that each consists of all algebras satisfying all the
identities on all its natural operations). If we take the algebraic category equationally
defined by the identities listed above, we get
Monoids with
commutation rule
3

full
Gr ∪ CommVVV
VVVV
VVVV
VVV
underlying VVVVVVVV+
set functor
Sets
*
/ Monoids
hh
h
h
h
hhhh
hhhh
h
h
h
ht hhh
where the descending dotted arrow is a faithful functor which however does not reflect
isomorphisms. That is, there exists a monoid (necessarily not satisfying the monomorphic
cancellation rule) on which there exist two different self actions satisfying the commutation
rule xa = axa . Of course, one interest for operations of this sort on a monoid C is the strong
op
properties it implies for the category S C of right actions on sets (or any Boolean topos
S) in particular with regard to the properties of the intrinsically defined “intuitionistic”
negation operator defined on the sub-actions A of any action X by
¬A = {x ∈ X| ∀r ∈ C[xr ∈
/ A]}.
TAKING CATEGORIES SERIOUSLY
9
Namely, if the monoid C admits a self-action with a commutation rule as above, then any
non-empty A contained in X = T (= the action of C on itself by right multiplication)
satisfies ¬¬A = T . By contrast, if C is so non-grouplike and non-commutative as the free
monoid on two generators, then every “principal” A = wC ⊂ T satisfies ¬¬A = A.
4. Functors from school mathematics mutate monoids into ordered sets,
and back
Before passing to the discussion of non-autonomous dynamical systems let us point out a
crucial example of a functor which occurs in school mathematics: suppose x ≤ y ≤ z are
non-negative integers or non-negative reals, then the differences y − x, z − y, z − x are
also non-negative and satisfy
z − x = (z − y) + (y − x)
0=x−x
Even though this theorem is a very familiar and useful identity, it cannot be explained
by either the monotonicity property nor as a homomorphism property in the usual sense,
for in fact it is a structure-preserving property of a process (namely difference) that goes
from a pre-ordered set to a monoid. As we have already pointed out, posets and monoids
are on the face of it very “opposite” kinds of categories, thus it appears that once we
have recognized the necessity for giving a rational status to something as basic as the
difference operation discussed above, we are nearly compelled to accept the category of
categories, since it is the only reasonable category broad enough to include objects as
disparate as posets and monoids and hence to include the above difference operator as
one of the concomitant structural mutations. To be perfectly explicit, let us denote the
relation x ≤ y (in the poset of quantities in question) by f , and similarly y ≤ z by g.
Define
φ(f ) = y − x
and similarly φ(g) = z − y and φ(1x ) = 10 for any x where 1x denotes x ≤ x. 0 may be
identified with (the identity morphism of) the unique object of the monoid of quantities
where composition is addition. Then φ satisfies
φ(gf ) = φ(g)φ(f )
φ(1x ) = 10
and hence is precisely a functor from a poset to a monoid. We can be still more explicit.
In our example, what does f : x ≤ y mean? We could identify f with the proof that
x ≤ y holds, that is with the non-negative quantity f such that x + f = y, or in other
10
F. WILLIAM LAWVERE
words with the morphism f in the monoid for which
0 ==
x
0
f
==y
==
=
/0
commutes. That is f is a morphism in the so called “comma category” 0/C where 0 is the
unique object of the monoid C. Of course, f is just the difference, so that φ is identified
as the forgetful functor
0/C
φ
C
well-defined for any comma category. Note that the comma category will be a poset only
in case C satisfies a cancellation law.
This construction can also be applied to P = multiplicative monoid whose morphisms
are positive whole numbers. Then 1/P is isomorphic to the poset whose objects are positive whole numbers ordered by divisibility, and under that identification, the functoriality
of the forgetful functor back to the monoid is expressed by
n|m & m|r =⇒
r m
r
=
· .
n
m n
There are non-trivial consequences of these observations, for any forgetful functor (such as
1/P −→ P) on a comma category satisfies the “unique lifting of factorizations” property:
φ(f ) = vu implies there are unique v̄, ū such that f = v̄ū, φ(v̄) = v, φ(ū) = u. If P is
a category satisfying a suitable local finiteness condition then we can define an algebra
structure on the set a(P) of all complex-valued functions on the set of morphisms of P
by the convolution formula
(β ∗ α)(f ) =
β(b)α(a).
ba=f
φ
Then the “unique lifting of factorizations” property of a functor P −−→ P is just what
is needed to induce a convolution-preserving homomorphism a(P) −→ a(P ). In case P
has cancellation, we thus get an inclusion of the Dirichlet algebra a(P) into the algebra
a(1/P) associated to a poset; in particular the µ-function (defined as the ∗-inverse, when
it exists, of the constantly 1 function) of P becomes the µ-function of the poset 1/P.
Since the ordering in 1/P is by divisibility, one thus sees how the functions µ, µ2 , etc. in
a(P) can be related to counting primes.
TAKING CATEGORIES SERIOUSLY
11
5. Non-autonomous systems, monoidal categories, Dedekind completeness,
and the construction of the real numbers as an abstraction of dynamical
waiting-times
Because right actions (= contravariant functors) of C are more directly related to the
analysis of C itself (for by the Cayley-Dedekind-Grothendieck-Yoneda lemma, there is a
op
canonical full embedding C → S C ) we are led to repeat the above discussion also for the
comma categories C/0 −→ C whose morphisms are commutative triangles
/•
•>
>>
>>
x >>
0

y
In case C is an additive monoid with cancellation, we would thus naturally write x ≥ y to
denote the existence of a morphism x → y in C/0. Since for any object X of a category
op
of the form S C (where S is the category of sets and C is any small category) there is an
equivalence of categories
op
op
S C /X ∼
= S (C/X)
where C/X is the “category of elements” of X, we get in particular for a commutative
and cancellative monoid C that
∼
S C /T −−→ S (C/0)
op
op
where T is C acting on itself on the right and C/0 is the poset described above. How
is the equation in the box to be interpreted in terms of dynamical systems? Well, T is
the simple autonomous dynamical system whose “states” reduce to just the instants of
time themselves, and an object of the left hand category is just an arbitrary autonomous
dynamical system X equipped with an equivariant morphism X −→ T ; (of course many
X, for example any periodic one, will not admit any such further structure X −→ T ).
The nature of the functor from left to right is just to consider the family of fibers (another
instance of a
construction) Xt of the given map X −→ T as t varies through T , and
whenever t ≥ t the global dynamics of X induces a map Xt −→ Xt , which completes
the specification of a functor (C/0)op −→ S corresponding to X. A natural interpretation
op
of the objects of S (C/0) is that they are non-autonomous dynamical systems, such as
arise from the solution of ordinary differential equations which contain “forcing” terms
or whose inertial or frictional terms depend on time in some manner (such as usury or
heating) external to the self-interaction modeled by the differential equation itself. In
general for a non-autonomous system the space of states Xt available at time t may itself
depend on t. The (left) adjoint functor is, as already remarked, actually an equivalence
of categories; it assigns to any non-autonomous system the single state space
Xt
t∈T
12
F. WILLIAM LAWVERE
with the single autonomous dynamics naturally induced by the non-autonomous dynamics
given as Xt −→ Xt for t ≥ t. In case all the instantaneous state spaces are identifiable
as a single Y
Xt ∼
= Y all t
then we see that the associated autonomous system is identifiable with
T ×Y
and this is just the universally-used construction for making a system autonomous: augment the state space by adding one dimension.
In case C is a commutative monoid, the category C/0 becomes a “symmetric monoidal”
category in the sense that there is a “tensor” functor
C/0 × C/0 −→ C/0
induced by the composition (which we will also write as +) and having the terminal
0
object ↓ of C/0 as “unit” object. In many cases this “tensor” has a right adjoint “Hom”
0
(C/0)op ×(C/0) −→ C/0 that can be naturally denoted by “subtraction”; it is characterized
(assuming C has cancellation) by the logical equivalence
t+a≥s
.
t≥s−a
In the fundamental example where C is the monoid of non-negative real (or rational)
numbers, the meaning of “subtraction” is forced by this adjointness to be truncated subtraction. This adjointness persists after completing, as discussed below.
h
For a poset (using ≥ for →) to be complete means that for any functor
M −−→ M ,
h ( ) exists in the poset (these are essentially infima) and also that h ( ) exists in the
poset (these are essentially arbitrary suprema). To “complete” the poset of non-negative
reals (or rationals) means roughly to adjoin (reals and) ∞, but since the precise meaning
of this in terms of one-sided Dedekind cuts is sensitive to the precise nature of the internal
cohesiveness/variation of the “sets” in S, it is fortunate that there is a precise analysis
of this process that goes back to the monoid C. Namely, define Pos → T to be the
intersection of all the subdynamical systems P of T which are large enough so that, given
any family f (p) ∈ T indexed by p ∈ P and satisfying f (p) + t = f (p + t) for all p ∈ P ,
t ∈ T , there exists a unique s ∈ T such that f (p) = s + p for all p ∈ P . Then in favorable
examples C is “continuous” (not necessarily complete) in the sense that
p ∈ Pos =⇒ ∃p1 , p2 ∈ Pos[p = p1 + p2 ]
and Pos itself is the smallest such P . It is then reasonable to consider the subcategory
op
i
A −−∗→ S C of “semicontinuous dynamical systems” defined to consist of those X for
13
TAKING CATEGORIES SERIOUSLY
f
which every “possible future” Pos −−→ X comes from a unique present state, or in other
words for which the inclusion Pos → T induces a bijection (T, X) −→ (Pos, X) between
the indicated sets of C-equivariant (= natural) morphisms. Then the inclusion i∗ has a left
op
i!
S C including the “identical” category A
adjoint i∗ that in turn has a left adjoint A −−→
op
as an “opposite” (to i∗ ) subcategory of S C . This implies that A is a topos whose truth
value object ΩA has as elements all the A-subobjects of T ; essentially the same is true
of R = A/i∗ T , the category of semicontinuous non-autonomous dynamical systems. But
def
its truth values are also the subobjects of the terminal object 1T because R is the topos
of sheaves on a topological space [0, ∞] topologized in such a way that there are as many
open sets as points, with ∞ corresponding to the empty set (or truth value “false”). A
dynamical system X in R has a real number |X| as its support, namely inf{t| Xt = 0},
and this construction can also be viewed as a functor
R −→ V
from the topos R to the poset V = [0, ∞] (the latter having ≥ as arrows) and both R, V
have tensor and Hom operations related to addition and truncated subtraction, that are
compared by R −→ V. Because this discussion can be viewed as a deep objective version
of Dedekind’s constructions of V, it appears that R as well as the A that gives rise to it
as a comma category, should be taken seriously.
6. Metric spaces as enriched categories clarify rotations
as “homotopy invariants”
It was extensively discussed in a 1973 seminar in Milan [8] that categories enriched in V
are just metric spaces and hence that a detailed mutual clarification of enriched category
theory [9] and metric space theory can be exploited. Continuing to take that remark
seriously between 1973 and the Bogotá meeting of 1983 led me to several additional
points of mutual clarification, that I will now explain. For a metric space A we have
0 = A(a, a)
A(a, b) + A(b, c) ≥ A(a, c)
in V
for any triple a, b, c of objects (points) of A; in general as explained in the cited article,
it is better, both for the theory and for the examples, not to insist on further axioms of
finiteness or symmetry. V-functors A −→ B turn out to be just distance-non-increasing
maps, and the V-object of V-natural transformations between two such f , g is easily
proved to be
B A (f, g) = sup B(f a, ga).
a∈A
φ
More general than V-functors are the V-modules (= V-relations = profunctors) A −−→ B
φ
which may be viewed as V-functors B op × A −−→ V; the composition of such arising from
14
F. WILLIAM LAWVERE
the enriched category notion of Trace (= “coend” = tensor product of modules) can be
shown in this case to reduce to
(ψ ◦ φ)(c, a) = inf [ψ(c, b) + φ(b, a)]
b∈B
φ
ψ
for A −−→ B −−→ C.
Note that if φ, ψ happened to have only ∞ = false and 0 = true as values, then ψ ◦ φ
would reduce to the usual ∃, ∧ composition of relations as a special case of the above
“least-cost” composition that arises when all of V = [0, ∞] is admitted. This relationship
can be made even more explicit as follows:
Let V0 be the two-object closed category false → true with conjunction as tensor and
logical implication as internal Hom. Then the inclusion V0 → V defined in the previous
paragraph is actually a closed functor that has a right adjoint V −→ V0 inducing the
poset structure on any metric space; that exemplifies the kind of de-enrichment process
which is a universal possibility in enriched category theory, giving the underlying ordinary
category for categories enriched in any closed category V. However, in our example there
is moreover also a left adjoint
π
V −−0→ V0
to the inclusion that is also a closed functor. [I have called it π0 because of the close
analogy with other graphs of adjoint functors
−−−−discrete
−−−−−→
←−−−
−−−−−− E0
points
−−−−−−−−−→
components
E
that occur, such as E = simplicial sets, E0 = sets, and because of the tradition in topology
of calling the components functor π0 because it is sometimes part of a sequence in which
the next term is the Poincaré groupoid π1 ]. Because π0 is closed, if we consider for any
metric space A the relation defined by
π0 A(a, b)
we get another (usually very coarse) preordering. This trivial construction is the key to
a pedagogical problem as follows:
I wanted to give, for a beginning course in abstract algebra, the basic example of a
normal subgroup and a quotient group.
Translations → Motions −→
−→ Rotations
(where each point of the underlying space should give rise to a splitting and hence to a
concrete representation of the abstract rotation group as a subgroup of motions, namely
those motions which are rotations about the point in question). However, it is desirable to
be able to make this basic construction before assuming detailed axioms on the structure
TAKING CATEGORIES SERIOUSLY
15
of the underlying metric space A. Motions should of course be defined as invertible Vfunctors f ; these will then in particular be distance-preserving
A(f a, f b) = A(a, b).
But how to define “translations”? At first is seems reasonable to say that they are motions
t which moreover satisfy A(ta, a) = constant, for it is not hard to show that the set of
such t is “normal”, i.e.
A(f tf −1 a, a) = A(tf −1 a, f −1 a) = same constant.
But they may not form a subgroup of the motions, for (theoretically) knowing nothing
about the structure of A, how could we know which constant should result from composing
t1 , t2 having constants c1 , c2 ? In this case the theoretical worry is substantiated by the
practical fact that we can construct an example of a five point metric space, indeed
embeddable in three-dimensional space as three equidistant points on the rim of a wheel
and two judiciously placed points on the axis through the center of the wheel, such that
the “translations” as defined are not closed under composition. But we shouldn’t give up.
Define a translation of an arbitrary metric space A to be an automorphism t such that
sup A(ta, a) < ∞.
a∈A
Now it is easy (assuming the metric is symmetric) to prove both parts of the statement that
the translations form a normal subgroup of the motions, as desired. Even better, there
are many examples of metric spaces A, such as the ordinary Euclidean plane, which have
the property that every translation does in fact move all points through the same distance
due to the “searchlight effect”: if A(ta, a) = A(tb, b) then A(tx, x) can be arbitrarily large,
for if there are enough strict translations we can assume that ta = a and construct
x
b
aX
X X
?
X tb
X X
?
X X
tx
Why is the left adjoint to the inclusion
{false, true} → [0, ∞]
the key to this problem? Because it (in contrast to the right adjoint, which seems to
admit as truly possible only those projects that cost no effort) is given by
π0 (s) = true iff s < ∞
as is easily verified. Thus it appears we should take seriously the idea that the homotopy
theory of metric spaces, that is the 2-functor
V-cat −→ V0 -cat
induced by π0 , is in large part the theory of rotations.
16
F. WILLIAM LAWVERE
7. Convexity, Isbell conjugacy, closed balls, and radii in enriched categories
The Cayley-Dedekind-Grothendieck-Yoneda embedding
A −→ V A
op
for V-enriched categories A, reduces in the case V = [0, ∞] under discussion to the fact
that A(−, a) is a distance-non-increasing real function for any point of any metric space
A, and that the sup-distance between two such functions is equal to the distance between
f
the given points or, more generally, that if Aop −−→ V is any distance-non-decreasing
nonnegative real function (not necessarily of the special form indicated) on the opposite
of the metric space A, then
A(a , a) + f a ≥ f a
all a
f a ≥ f a − A(a , a)
all a
f a ≥ V(A(a , a), f a )
all a
f a ≥ sup V(A(a , a), f a )
a
f a ≥ V A (A(−, a), f )
op
but the last inequality is actually an equality because the sup is achieved at a = a.
Now in between we can insert the space of closed subsets of A
A → F(A) → V A
op
by assigning to each F → A the function
F (a ) = inf A(a , a)
a∈F
which vanishes on (by definition) the closure of F . The sup metric on V A , restricted to
F(A) is a refined non-symmetric version of the Hausdorff metric; that is, its symmetrization is the Hausdorff metric, but it itself reduces via V −→ V0 to an ordering which reflects
the inclusion of the closed sets
op
F1 ⊆ F2
Since any f ∈ V A
op
as closed sets iff F1 ≥ F2
in V A .
op
does have a zero-set, therefore a right adjoint to the inclusion
F(A)
u

Z
/ V Aop
can be constructed. This leads to the idea that the objects in V A might be considered as
“refined” closed sets; a point of view already forced upon researchers in constructive analysis and variational calculus by the stringent requirements of their proofs and calculations
thus receives also a conceptual support from enriched category theory.
op
TAKING CATEGORIES SERIOUSLY
17
Now an extremely fundamental construction in enriched category theory is the adjoint
pair known as Isbell conjugation
V
Aop
o
( )∗
(
/
(V A )op
)#
which is defined in both directions by a similar formula
ξ ∗ (a) = V A (ξ, a)
op
α# (a) = V A (α, ā)
where we have followed the usual practice of letting the Yoneda lemma justify the abuses
of notation
A(−, a) = a in V A
op
A(a, −) = ā in V A .
The general significance of this construction is somewhat as follows: if V is the category of
sets, or simplicial sets, or (properly construed) topological spaces, or bornological linear
spaces, and if the V-category A is construed as a category of basic geometrical figures,
op
then V A is a large category which includes very general geometrical objects that can be
probed with help of A, but that would inevitably come up in a thorough study of A itself.
On the other hand, V A includes very general algebras of quantities whose operations (à
la Descartes) mirror the geometric constructions and incidence relations in A itself. Then
the conjugacies are the first step toward expressing the duality between space and quantity
fundamental to mathematics: ( )∗ assigns to each general space the algebra of functions
on it, whereas ( )# assigns to each algebra its “spectrum” which is a general space. Of
course neither of the conjugacies is usually surjective; the second step in expressing the
fundamental duality is to find subcategories with reasonable properties that still include
the images of the conjugacies:
X _ ho
V
z
zz
z
zz
|z
z
Aop
o
A GG
GG
GG
GG
G#
/ op
6 A _
/
A op
(V )
For example if V is the category of sets and A is a small category with finite products,
we could take X to be the topos of “canonical sheaves” on A, and A to be the algebraic
category of all finite-product-preserving functors.
Our poset V = [0, ∞] appears rather puny compared to the grand examples mentioned
in the previous paragraph, but seriousness eventually leads us to try the Isbell conjugacy
18
F. WILLIAM LAWVERE
on it as well, and in particular to ask which closed sets F ∈ F(A) are fixed by the
composed Isbell conjugacies for a metric space A.
F(A) u
( )∗#
Z
/ V Aop o
( )∗
(
/
(V A )op .
)#
Note that from the adjointness, ξ ≥ ξ ∗# for all ξ in V A , so that the idempotent operation
( )∗# gives a kind of lower envelope for functions and hence a kind of hull Z(ξ) ⊆ Z(ξ ∗# )
for the corresponding closed zero-sets; the question is what kind of hull?
To answer this it is relevant to explicitly introduce the following parameterized family
op
of special elements of V A :
op
B
V op ⊗ A −−→ V A
op
defined by
B(r, c)(a ) = V(r, A(a , c)),
where V(x, y) = y − x is the truncated subtraction in our example V = [0, ∞]; but for any
closed category V denotes its internal Hom, and where the tensor product of V-categories
(defined in our case as the metric which is the sum of the coordinate distances) is used
instead of the cartesian product because it guarantees that B itself is also a V-functor.
The letter B stands for closed ball of a given radius and center since
0 ≥ B(r, c)(a ) iff r ≥ A(a , c).
An amazing example of the seriously-pursued study of the mutual relationship of a
key example with general philosophy is that these “closed balls” occur and are useful over
many apparently quite diverse
in homological algebra.
closed categories V, for example
Aop
Moreover, let us denote by the supremum operation on V
ξi (a ) = sup ξi (a )
i
since that is what it corresponds to under the operation Z of taking zero-sets.
With the above-introduced notation, we see that ξ = ξ ∗# iff ξ ∗# ≥ ξ, and also that
α# (a ) = V A (α, ā )
= sup V(α(a ), A(a , a ))
a
so that
α# =
B(α(a ), a )
a
is the intersection of all the closed balls, centered at all points a , of specified radius α(a ).
[Naturally this construction, under the name of “end” or “center”, also comes up for
TAKING CATEGORIES SERIOUSLY
19
general V]. Now what if α is of the form ξ ∗ ? The number ξ ∗ (a ) is a radius (about an
arbitrary center a ) that ξ somehow prefers:
ξ ∗ (a ) = sup V(ξ(a), A(a, a ))
a
so that
r ≥ ξ ∗ (a ) iff r + ξ(a) ≥ A(a, a ) for all a
iff ξ(a) ≥ A(a, a ) − r for all a
iff ξ ≥ B(r, a ).
Thus (since throwing some larger balls into the family won’t change the intersection)
ξ ∗# = {B(r, a )| ξ ≥ B(r, a )}
is a geometrical description of the double-conjugate lower envelope of ξ. In the case
op
where ξ is fixed under the other idempotent operation on V A coming from Z, i.e. if ξ is
determined by its closed zero-set F as explained before
ξ(a ) = inf A(a , a)
a∈F
then the condition ξ ≥ B(r, a ) reduces to
∀a ∈ F ∀a
i.e. to
A(a , a) ≥ B(r, a )(a )
= A(a , a ) − r
A(a , a) + r ≥ A(a , a ) for a ∈ F, arbitrary a .
In particular this means that F ⊆ the ball (= zero set of) B(r, a ), so we see that
F ⊆ F ∗# ⊆ the intersection of all closed balls that contain F .
But in fact that intersection of balls is equal to F ∗# , since the right adjointness of Z says
in particular that for any closed set F and any ball we have the equivalence
F ⊆ ZB(r, a )
.
F ≥ B(r, a )
Putting it differently, since we have in general that
A(a , a) + A(a, a ) ≥ A(a , a ),
if a ∈ F implies 0 ≥ B(r, a )(a), i.e. r ≥ A(a, a ), then for any a and any a ∈ F
A(a , a) + r ≥ A(a , a) + A(a, a ) ≥ A(a , a )
20
F. WILLIAM LAWVERE
so that
meaning that
A(a , a) ≥ B(r, a )(a )
F (a ) ≥ B(r, a )(a ) as functions of a ,
because the right hand side
is independent of a and the function F was defined as the
infimum (adjointness of “ along a diagonal”). Thus
F ∗# = intersection of all closed balls which contain F
= closed convex hull of F ,
at least in many metric spaces of geometric importance, and for the others the proposed
use of the term “convex” for those closed sets F satisfying F = F ∗# should be as good
or better than other proposals because of the apparent importance of adjointness in calculations.
As noticed above, the numbers ξ ∗ (a ) have in many connections the concrete signifinf
icance of radii for balls about a . Using the obvious “direct limit” functor V A −−−→ V
(which exists because A −→ 1 is a V-functor), we can define a single “radius” for ξ itself
by applying the composite
VA
op
( )∗
/ (V A )op
inf op
/5 V op
rad
In particular, the radius of a closed set F ∈ F(A) ⊆ V A
op
is
rad(F ) = inf{r| ∃a [F ⊆ ZB(r, a )]}
where the candidate centers a are not themselves necessarily in F . (Note that to say we
have a functor F(A) −→ V op from a poset to V op is to say that the values increase as the
objects in F(A) increase). The radius is a more functorial quantity than the habituallyused “diameter”; for a symmetric metric space there is the estimate diam ≤ 2 rad
whereas for certain reasonably well behaved spaces one may also have a converse estimate rad ≤ diam. There is a strong tendency for the radius to be realized at a unique
center a . Note that rad(F ∗# ) = rad(F ).
8. Geodesic remetrization as an adequacy comonad
Now let us say a few words about the important role of paths in metric spaces. The
comma categories d/V = [0, d] with their canonical d/V → V have a retraction given
again by “double dualization”:
x −→ V(V(x, d), d) = d − (d − x)
TAKING CATEGORIES SERIOUSLY
21
is always in the interval; moreover single d-dualization is an invertible duality
(d/V) o
/
(d/V)op
when restricted, provided d < ∞. Denote by V ⊂ V all those d such that π0 (d) = true,
i.e. for which d < ∞. Let V(d) be the symmetric metric space determined by d/V, so
that in particular d − ( ) becomes (for d < ∞) a self-motion of the interval V(d), because
symmetrizing is a functor. Indeed, d −→ V(d) defines a functor
V op −→ V-cat
using d ≥ d =⇒ V(d) ⊆ V(d ), but also a functor V −→ V-cat by invoking the retractions.
The essential question we want to understand is: to what extent is the structure of an
arbitrary metric space A ∈ V-cat analyzable in terms of the paths (= V-functors)
V(d) −→ A
of duration d, with d variable? Since all constants are paths, such analysis easily maintains
the points of A. If there is such a path, passing a0 at time 0 and a1 at time d, then
d ≥ A(a0 , a1 ).
This leads to the idea of geodesic distance:
(ΓA)(a0 , a1 ) = inf{d ∈ V | ∃ σ : V (d) −→ A with σ(0) = a0 , σ(d) = a1 }
which can be seen to be a new metric since
/ V(d )
V(0)
V(d) 
/ V(d + d )
is a pushout in V-cat. That is, if σ is a path in A of duration d, and σ of duration d ,
and if
σ(d) = σ (0) = a1
we must show that if d ≥ t, d + d ≥ s ≥ d, then
?
A(σ(t), σ (s − d)) ≤ s − t.
But we have
A(σ(t), σ (s − d)) ≤ A(σ(t), a1 ) + A(a1 , σ (s − d))
= A(σ(t), σ(d)) + A(σ (0), σ (s − d))
≤d−t+s−d=s−t
22
F. WILLIAM LAWVERE
σ ∗σ
because d ≥ t and s ≥ d. The other cases of the V-functorality of (d + d ) −−−−→ A are
obvious. Thus (using the logicians’ symbol for “proves”)
σ d ≥ ΓA(a0 , a1 )
=⇒ σ ∗ σ d + d ≥ ΓA(a0 , a2 ).
σ d ≥ ΓA(a1 , a2 )
Because direct limits in metric spaces are essentially computable in terms of direct limits
of sets and infima of distances, it can be seen that
ΓA = lim V(dom σ)
−→
σ∈V /A
is an endofunctor of V-cat having a natural distance-non-increasing map
ΓA
A
back to the identity functor; it is in fact the “adequacy comonad” of V −→ V-cat, a
notion defined for any small subcategory of any category having direct limits. Note for
example that path-connectedness of A becomes π0 -connectedness of ΓA, for if two points
of A are not connectable by a path, then (empty inf) their geodesic distance in ΓA is
infinite.
9. History still has much to teach, and raises fresh questions
Finally, I believe that we should take seriously the historical precursors of category theory, such as Grassmann, whose works contain much clarity, contrary to his reputation for
obscurity. For example, I read there a statement of the sort “diversity can be added”
whereas “unity can be multiplied” together with quite convincing geometrical and algebraic substantiation of these principles. The first of them suggests the following:
If F(A) is a poset of parts of a space and C is a suitable additive monoid, then the
amount by which F must be extended to achieve a diverse G ⊇ F might be given by
µ(F, G) ∈ C; that should again give a functor F(A) −→ C in that
F ⊆ G ⊆ H =⇒ µ(F, H) = µ(F, G) + µ(G, H).
'
$
' $
F
& %
&
%
TAKING CATEGORIES SERIOUSLY
23
Of course, if 0 ∈ F(A) and if C has cancellation then µ(G, H) will be determined by µH =
def
µ(0, H) and µF . To further express the “quantitative” nature of such a measurement of
extension µ, we can consider the further condition that µ “only depends on the difference”.
Since the crucial property of difference is again that it is a Hom, adjoint this time to union
as ⊗,
S ⊇H \G
G∪S ⊇H
we can express this invariance of the functor µ using only the union structure on F (A):
{S| G ∪ S ⊇ H} = {S| G1 ∪ S ⊇ H1 } =⇒ µ(G, H) = µ(G1 , H1 ).
It would be interesting to determine for which upper semilattices F, 0, ∪ there exists
a commutative monoid C and a functor µ with this invariance property which moreover
has the “unique-lifting-of-factorizations” property previously discussed, i.e., for which the
object X in a process of intermediate expansion F ⊆ X ⊆ H is uniquely determined by
sufficiently many quantitative measurements of its size. For example, area alone is not
sufficient but C can be a cartesian product of many different kinds of quantities. Again
adjointness makes at least an initial contribution to the problem: for each F there is a
well-defined universal C and µ, so that one need only study the lifting question for that.
10. Dialectical relation between teaching and research can be exemplified
by metrical development of enriched category theory
We have seen that the application of some simple general concepts from category theory
leads from a clarification of basic constructions on dynamical systems to a construction of
the real number system with its structure as a closed category; applied to that particular
closed category, the general enriched category theory leads inexorably to embedding theorems and to notions of Cauchy completeness, rotation, convex hull, radius, and geodesic
distance for arbitrary metric spaces. In fact, the latter notions present themselves in such
a form that the calculations in elementary analysis and geometry can be explicitly guided
by the experience that is concentrated in adjointness. It seems certain that this approach,
combined with a sober appreciation of the historical origin of all notions, will apply to
many more examples, thus unifying our efforts in the teaching, research, and application
of mathematics.
References
[1] Eilenberg, S. and Mac Lane, S., General Theory of Natural Equivalences. Trans.
Amer. Math. Soc. 58(1945), 231–294
[2] Grothendieck, et. al., Theorie des Topos et Cohomologie Etalé des Schemas. Springer
Lecture Notes in Mathematics 269, 270, 1972.
24
F. WILLIAM LAWVERE
[3] Lawvere, F.W. (editor), Toposes, Algebraic Geometry and Logic. Springer Lecture
Notes in Mathematics 274, 1972.
[4] Lawvere, Maurer, and Wraith (editors), Model Theory and Topoi. Springer Lecture
Notes in Mathematics 445, 1975.
[5] Banaschewski (editor), Categorical Aspects of Topology and Analysis. Springer Lecture Notes in Mathematics 915, 1982.
[6] Lawvere and Schanuel (editors), Categories in Continuum Physics. Springer Lecture
Notes in Mathematics 1174, 1986.
[7] Joyal, A., Une théorie combinatoire de séries formelles, Advances in Mathematics,
42(1981), 1–81.
[8] Lawvere, F.W., Metric Spaces, Generalized Logic, and Closed Categories, Seminario
Matematico e Fisico di Milano 43(1973), 135–166, and Reprints in Theory and Applications of Categories, no. 1(2002), 1–37.
[9] Kelly, G.M., Basic Concepts of Enriched Category Theory. London Mathematical
Society Lecture Notes Series 64, Cambridge University Press, 1982 and Reprints in
Theory and Applications of Categories, (2005).
University at Buffalo, Department of Mathematics
244 Mathematics Building, Buffalo, New York 14260-2900
Email: wlawvere@buffalo.edu
This article may be accessed from http://www.tac.mta.ca/tac/reprints or by anonymous ftp at
ftp://ftp.tac.mta.ca/pub/tac/html/tac/reprints/articles/8/tr8.{dvi,ps}
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Robert Paré, Dalhousie University: pare@mathstat.dal.ca
Jiri Rosicky, Masaryk University: rosicky@math.muni.cz
Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu
James Stasheff, University of North Carolina: jds@math.unc.edu
Ross Street, Macquarie University: street@math.mq.edu.au
Walter Tholen, York University: tholen@mathstat.yorku.ca
Myles Tierney, Rutgers University: tierney@math.rutgers.edu
Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it
R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca